Summary: For Lucas sequences of the first kind $ (u_n)_{n\ge 0}$ and second kind $ (v_n)_{n\ge 0}$ defined as usual by $ u_n=(\alpha ^n-\beta ^n)/(\alpha -\beta ), v_n=\alpha ^n+\beta ^n$, where $ \alpha $ and $ \beta $ are either integers or conjugate quadratic integers, we describe the sets $ \{n\in\mathbb{N}:n$ divides $ u_n\}$ and $ \{n\in\mathbb{N}:n$ divides $ v_n\}$. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called $basic$ number, which can only be $ 1, 6$ or $ 12$, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.

11B39

Lucas sequences, indices